Sunday, May 27, 2018

From some time now, I have been immersing myself into the world of Improv. It is a great practice and it is not only a good distraction, but it also provides some great life lessons.

One of the important skills in Improv is to be in the moment. Several games in improv emphasize this, and one of them is called Enemy/Defender or Assassin.

In a group of people, everyone secretly chooses an enemy, whom they have to avoid, and a defender, whom they need to keep between them and their enemy. This defines the game dynamics. From a mathematical perspective, this defines a two dimensional dynamical system.

Each person is represented by a point. Each point has an enemy and a defender. Each point has to move in order to put its defender between them and their enemy. This can be modeled by $n$ points, with $n\geq 3$. The positions of the points can be taken to be vectors $P_i=(x_i,y_i)$. Associated with each $i$, there is an enemy and a defender $j$ and $k$, with $j\neq i\neq k$, with position vectors $E_i$ and $D_i$ respectively.

The goal then can be stated as having $D_i$ between $P_i$ and $E_i$, for all $i$. The condition of motion depends whether $P_i$ is on the ray $\overrightarrow{E_iD_i}$. This can be measured by the distance between $P_i$ to the line $E_iD_i$ and the projection of $\overrightarrow{P_iD_i}$ onto $\overrightarrow{E_iD_i}$.

Then, the dynamics of the system can be obtained by moving each point $P_i$ in order to satisfy these conditions on small time frames. It is important to avoid collisions and to restrict the $P_i$ to be only in the room to represent a more realistic situation.

This is a simulation in which for every $P_i$, a change in position $v_i$ is computed in terms of all the positions $\{P_1,P_2, \dots, P_n\}$ together with $E_i$ and $D_i$. Here, the initial positions and the enemy/defender arrangement is randomly generated, giving different behaviors depending on the initial configurations. The dynamics also depend on the parameters of the $v_i$, such as size and direction. Just changing these by a little could change the entire dynamics for the system. This is to be expected, as the stability of orbits are very sensitive to initial data (eg. Irrational rotations, Billiard dynamics, etc.)

Overall, Improv showed to be a great activity and even a nice excuse to use dynamical systems. Being in the moment and stable sometimes mean to avoid irrationality and incommensruable lengths. (Pun intended)